Bounded scalar functions on an index set
= Bounded scalar functions on an index set
{title2=$\ell_\infty(\Gamma)$}
For an arbitrary set $\Gamma$ and scalar field $\mathbb F$, this is the space of bounded functions $u:\Gamma\to\mathbb F$ with the <supremum norm> $\|u\|=\sup_\gamma|u(\gamma)|$. Uniform limits of Cauchy sequences prove it is a <Banach space>. For $\Gamma=\mathbb N$ it is the <l-infinity sequence space>; for the empty index set it is the zero space, with norm zero.